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(w, a) ==/*,    V

154       XIII    INTEGRATION
each integer m ;> 1 we have

(same method).

(e)   Put k(u) = sup h(u, oc), where the supremum is taken over all finite partitions of

at

X into u-integrable subsets. The number h(u) is called the entropy of u (relative to
the measure /*). Show that, for each integer m ^ 1, we have h(um) = mh(u). (Remark
that

h(um, a) g h(um, V w~J(a) ) = w/z(w, a).)
\      '-           /

If moreover u is bijective and u"1 is ^-measurable, then hfa-1) = h(u).

(f)   Show that A(lx) = 0, where lx is the identity mapping of X. Deduce that if

u* = ix for some integer p > 0, then h(u) = 0.

10. INTEGRALS OF VECTOR-VALUED FUNCTIONS

Consider first a mapping f of X into a/w'/tf-dimensional real vector space

m

E. Let (ei)1^f^m be a basis of E, and put fto^^ffcfa for each xeX.

Then f is said to be integrable (with respect to JJL) or ^integrabk if each of the
real- valued functions ft is integrable, and we define

(13.10.1)

f
J

It is immediately checked that this definition is independent of the basis
chosen. If ||z|| is a norm defining the topology of E, then we have the follow-
ing criterion :

(13.10.2)   A mapping f : X ->E is integrable if and only iff is measurable and
J *j|f(x)|| djjt(x) < + oo. The function x\-^> ||f(x)|| is then integrable.

For f is measurable if and only if the ft(l ^i^ri) are measurable, by
(13.9.6); also there exist two constants a, b such that

by (5.9.1). The result now follows from (13.9.13) and (13.9.6).

In particular, a complex function/ on X is integrable if and only if+n-l            \